A new approach to entropy production minimization in diabatic distillation column with trays

A NEW APPROACH TO ENTROPY PRODUCTION MINIMIZATION INDIABATIC DISTILLATION COLUMN WITH TRAYS

by

Mom~ilo Dj. SPASOJEVI] a, Milovan R. JANKOVI] b, and Damir D. DJAKOVI] a*

a Fac ulty of Tech ni cal Sci ences, Uni ver sity of Novi Sad, Novi Sad, Ser biab Fac ulty of Tech nol ogy, Uni ver sity of Novi Sad, Novi Sad, Ser bia

Orig i nal sci en tific pa perUDC: 66.045.1:66.021.3.048:519.876.5

DOI: 10.2298/TSCI1002317S

Pre vi ous ap proach to di rect nu mer i cal minimization of en tropy pro duc tion indiabatic dis til la tion col umn in or der to de ter mine heat quan tity to be ex changed attrays was based on tem per a tures on trays as con trol vari ables and it was ap pliedonly to sim ple bi nary col umns. Also, pre vi ously de vel oped the o ret i cal mod els forde ter min ing op ti mal ex changed heat pro file were de ter mined only at such col umnsand while they were ap prox i mated they pro duced worse re sults than nu mer i calmin i mum of en tropy pro duc tion. In this pa per, as con trol vari ables forminimization, ex changed heat on the trays is used. It en ables ap pli ca tion to com -plex multicomponent diabatic col umns. Ishii-Otto global method, based on modellinearization and it er a tive so lu tion by New ton-Raphson tech nique, is ap plied forsolv ing col umn math e mat i cal model. Needed thermodynamical prop er ties for idealsys tems are cal cu lated us ing Lewis-Randall ideal so lu tion model, and for non-ideal slightly po lar sys tems they are cal cu lated us ing Soave equa tion of state. Five di rectmeth ods are used for nu mer i cal op ti mi za tion. Ap plied ap proach is suc cess fullydem on strated at fre quently used ex am ple of dis til la tion of ben zene and toluol mix -ture by us ing for these pur poses spe cially writ ten pro gram. Sim plex method ap -peared to be the most con ve nient op ti mi za tion method for the con sid ered prob lem.

Key words: entropy production, diabatic distillation, optimization

In tro duc tion

Dis til la tion is still the most fre quently used sep a ra tion tech nique al though it has smallen ergy ef fi ciency. Dur ing dis til la tion, heat en ergy de grades and heat from higher tem per a turelevel shifts to a lower one while some fuel is spent in or der to bring heat en ergy to a needed level. In the last 50 years, ef forts have been made to in crease dis til la tion en ergy ef fi ciency throughheat in te gra tion of both other pro cesses (exchangers net work, cou pled col umns) and the dis til la -tion col umn by it self (heat pumps, heat in te grated dis til la tion col umn – HIDIC). In the last 35years, a so called diabatic dis til la tion has been in ves ti gated as one of pos si bil i ties. In tra di tional(adi a batic) dis til la tion col umn with trays, heat is in tro duced in a reboiler, and re moved in a con -

Spasojevi}, M. Dj., et al.: A New Approach to Entropy Production Minimization in Diabatic ...

THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp. 317-328 317

* Corresponding author, e-mail: [email protected]

denser while in a diabatic col umn, heat is ex changed at ev ery tray (it is in tro duced in a strip pingand re moved in a rec ti fy ing sec tion) by in de pend ent (in ter me di ate heat exchanger for ev erytray) or se quen tial exchangers (with an exchanger for the strip ping and rec ti fy ing sec tions) in or -der to min i mize en tropy pro duc tion and exergy loss, re spec tively, which max i mizes ef fi ciencyin ac cor dance with the Sec ond Law of Ther mo dy nam ics. Bor der line case of diabatic dis til la tionis re vers ible dis til la tion (with a con tin ual heat ex change) which is de scribed, e. g. in [1-3]. Suchcol umn has in fi nite length (tray num ber) and is only of the o ret i cal in ter est. More prac ti cal ap -proach is ad di tion of one or sev eral exchangers in the col umn in or der to de crease ir re vers ibil ity. The ef fects and con di tions of ap pli ca tion of lim ited num ber of exchangers are in ves ti gated,among oth ers, in [4-9], by the use of dif fer ent meth ods. Minimization of en tropy pro duc tion indiabatic col umn with the exchangers at ev ery tray has been more in ten sively in ves ti gated in thelast 15 years and fol low ing meth ods are sug gested:– equipartition of entropy production [10],– equipartition of forces (EoF) [5, 11],– equal thermodynamic distance (ETD) [12, 13], and– direct numerical optimization by different methods.

Equal dis tri bu tion of driv ing forces is the o ret i cally de rived method which has arisen as the ex pan sion of method of equal dis tri bu tion of en tropy pro duc tion on ir re vers ible ther mo dy -nam ics ba sis. This method pro poses that gas eous and liq uid stream ex chang ing heat and masson a col umn tray should work with uni form driv ing force through in ter fa cial area. Equal ther mo -dy namic dis tance method, also the o ret i cally de rived for bi nary dis til la tion, or ders that for ev erypair of con sec u tive trays there should be equal dis tance re gard ing Weinhold met rics [14] in thespace of ther mo dy namic vari ables. The re sults of ap pli ca tion of sug gested meth ods and theirmu tual com par i son are given in sev eral pa pers, among oth ers, in [15-25].

Pre vi ous in ves ti ga tions and ap pli ca tion of sug gested meth ods, both the o ret i cal andnu mer i cal, have been lim ited to sim ple bi nary col umns. The o ret i cal meth ods give only ap prox i -ma tion of op ti mal ex changed heat pro file, i. e., they are more in fe rior than nu mer i cal op ti mi za -tion.

Prob lem state ment

Dis til la tion in sim ple bi nary col umns rep re sents only small share of dis til la tion in to -tal, so there is a need to de velop a pro ce dure for en tropy pro duc tion minimization, which shouldbe ap plied at com plex multicomponent diabatic dis til la tion col umns. Pre vi ous ap proach ofminimization was based on tem per a tures as con trol vari ables. This ap proach can be ap plied tocom plex col umns in two ways. The first way is that minimization of en tropy pro duc tion rep re -sents op ti mi za tion in pres ence of lim its given by equa tions of math e mat i cal model of the col -umn, ex cept equa tions of en ergy bal ance from which needed quan ti ties of ex changed heat arecal cu lated. This prob lem is dif fi cult to solve be cause of ex pressed non-lin ear ity and nar rowrange of vari ables in which it is pos si ble to cal cu late needed fluid char ac ter is tics. It also re quires com plex so phis ti cated op ti mi za tion meth ods. In the sec ond way, solv ing pro ce dure of col umnmodel the equa tions, ex cept en ergy bal ance equa tions, will be done in or der to de ter mine quan -ti ties which are needed for goal func tion cal cu la tion – en tropy pro duc tion. This will en able ap -pli ca tion of sim pler op ti mi za tion meth ods but it will also de mand de vel op ment of new al go -rithm for model equa tions solv ing.


318 THERMAL SCIENCE: Year 2010, Vol. 14, No. 2, pp. 317-328

In this pa per, the third ap proach is ap plied to con trol vari ables for op ti mi za tion. In -stead of tem per a tures ex changed heat quan ti ties at the trays are used as con trol vari ables. Thischoice en ables ap pli ca tion of sim pler di rect op ti mi za tion meth ods, and for solv ing the wholecol umn model, it is pos si ble to ap ply some of nu mer ous well known and checked meth ods. Col -umn mod els, so lu tion, and op ti mi za tion ap proaches in pre vi ous use, and also in new sug gestedpro ce dure are given.

A model of sim ple bi nary col umn with ideal trays

Sim ple adi a batic bi nary col umn with ideal trays and to tal con denser, shown in fig. 1,serves for bi nary mix ture sep a ra tion and in cludes feed mix ture (F) and two prod ucts – dis til late(D) and bot tom prod ucts (B). The col umn is equipped with a con denser at the top (to tal) and areboiler at the bot tom. An anal o gous diabatic col umn with the exchanger at ev ery tray is shownin fig. 2. Al though a re flux into the col umn does not come back from the con denser at the top, itserves only for va por con den sa tion from the top into dis til late.

Diabatic col umn can be de scribed by fol low ing equa tions of ma te rial and en ergy bal -ances and us ing the equa tions of equi lib rium and sum ma tion:

F = D + B (1)



Fig ure 1. Sim ple adi a batic col umn withto tal con denser

Fig ure 2. Sim ple diabatic col umn with to tal con denser

V LD n n

B n n Nn n

F

F+ - =

Î -

- Î

ìíî

1

0 1[ , ]

[ , ](2)

Fz Dx Bx kk k k= + ÎD B [ , ]1 2 (3)

V y L xDx

Bx

n n

n n Nkn k n n k n

k

k

F

F+ + - =

-

Î -

ÎÎ1 1

0 11 2

D

B

[ , ]

[ , ][ , ]

ìíî

(4)

Q D H H0 1= -( )D V (5)

Q V H L H V H L H nn n n n n n n n n= + - - - Î+ + - -

V L V L1 1 1 1

1 2c [ , ] (6)

c ==ì

íî

FH n nF

otherwise0

F

Q V H BH L HN N N N N+ + += + -1 1 1

V B L (7)

yk n = Kk nxk n kÎ[1, 2] nÎ[1, N + 1] (8)

x y n Nk k

k n k n= =å = å = Î +

1

2

1

21 1 1 1[ , ] (9)

En tropy pro duc tion in the con denser, col umn trays, and the reboiler is:

d

d

irrD VS

tDS V S

Q

T

æ

èç

ö

ø÷ = - -

0

1 10

0

(10)

d

d

irrL V L nS

tV S L S V S L S

Q

Tn

n nV

n n n n n n

æ

èç

ö

ø÷ = + - - -+ + - -1 1 1 1

n

- Îk n N[ , ]1 (11)

k ==ì

íî

FS n nFF

otherwise0

d

d

irrV B

NLS

tV S BS L S

Q

TN

N N NN

N

æ

èç

ö

ø÷ = + - -

+

+ ++

+1

1 11

1

(12)

On the ba sis of ex pres sions (10)-(12), to tal en tropy pro duc tion in the col umn is:

d

d

d

d

irr irrD B FS

t

S

tDS BS FS

Q

Tn

n

nn

N

n

n

=æ

èç

ö

ø÷ = + - -

=

+

=å

0

1

0

+

å1

(13)

If en tropy pro duc tion in the exchangers is taken into ac count, to tal en tropy pro duc tionwill be [21]:

d

d

irrD B F HXS

tDS BS FS

Q

TQ Xn

nn

N

n nn

n

= + - - +=

+

=

+

å å0

1

0

1

(14)



Driv ing force for heat ex change in the exchangers, X nHX , can be ap prox i mated in the

fol low ing way [21, 23, 24]:

XQ

T An

n

n n n

HX =d

l 2(15)

Ap pli ca tion of eqs. (14) and (15) is im por tant when op ti mal dis tri bu tion of exchangerarea, An, is searched for the fixed to tal area of all exchangers.

In pre vi ous prac tice of en tropy pro duc tion in the col umn minimization, eq. (13) or eq.(14), re spec tively, in the case of in de pend ent exchangers at trays, tem per a tures from T2 to TN

(T0, T1, and TN+1 are fixed by given com po si tion of dis til late and bot tom prod ucts) are used ascon trol vari ables. A solv ing pro ce dure and de ter mi na tion of ex changed quan ti ties of heat attrays is:

– mo lar frac tions of heavier com po nent are ex pressed from the equa tions of sum ma -tion, eq. (9):

x2 n = 1 – x1 n y2 n = 1 – y1 n

and in tro duced into equi lib rium, eq. (8):

y K x

y K x

n n n

n n n

1 1 1

1 2 11 1

=

- = -( )(16)

Ob tained sys tem of two equa tions with two un knowns is lin ear if it is an ideal and con -stant va por – liq uid equi lib rium, Kk n is de pend ant only on tem per a ture and pres sure, re spec -tively, and non-lin ear if the sys tem is non-ideal and the con stant is de pend ent both on liq uid andva por phases com po si tion. The sys tem is solved for all trays and in that way, va por and liq uidphase com po si tion are de ter mined.– From total material balance equation, eq. (2), is expressed Vn+1 and introduced in lighter

component material balance equation, eq. (4), and then Ln, and Vn+1 from the expression (2)are calculated for all trays.

– On the basis of known composition and temperatures vapor and liquid streams, enthalpiesand then exchanged heat on the trays from energy balance equations, eqs. (5)-(7), and at theend, values of goal function, given by eqs. (13) or (14) are calculated.

– In accordance to the chosen optimization method, temperature profile on trays is changedand solving procedure is repeated until minimum goal function, given by eqs. (13) or (14) isestablished.

In the case of col umn with se quen tial heat exchangers, con trol vari ables are the flowsand in put tem per a tures of cold and hot flu ids and exchanger sur face dis tri bu tion on trays. Dur -ing op ti mi za tion for ev ery set of con trol vari ables, tem per a ture pro file which cor re sponds to asta tion ary state is searched [22], while part of pre vi ously de scribed cal cu la tion for com po si tionis used for flow and va por and liq uid streams enthalpies de ter mi na tion in the col umn. Dif fer entnu mer i cal meth ods are used for the op ti mi za tion of diabatic col umn with in de pend ent and se -quen tial heat exchangers:– Powel method [17, 19, 20], – optimization through small perturbations [18],– Monte Carlo method [19, 21],– simplex method [8, 22]



– least squares regression with Gauss-Newton procedure and one-dimensional search [23],and

– sequential quadratic programming using fmincon Matlab functions [21, 24, 25].The use of tem per a tures on trays as con trol vari ables and de scribed way of solv ing ma -

te rial and en ergy bal ances in or der to de ter mine op ti mal pro file of ex changed heat re strict ap pli -ca tion to bi nary mix tures and sim ple col umns only.

More log i cal choice is to take con trol vari ables to be the quan ti ties of heat ex changedon trays, so for solv ing col umn model equa tion, some of nu mer ous known pro ce dures for com -plex multicomponent col umns can be used. In that case, even choice of op ti mi za tion method isnot a crit i cal one.

Com plex multicomponent model with idealtrays

The tray of com plex col umn for multicomponentmix ture sep a ra tion on the ba sis of va por-liq uid equi -lib rium is sche mat i cally shown in fig. 3. At ev ery tray, feed mix ture and re moved va por and liq uid side prod -ucts and ex changed heat can be in tro duced.

Math e mat i cal col umn model with such trays is de -scribed by fol low ing equa tions:

– equa tion of to tal ma te rial bal ance

con denser V1 – L0 – SL0 – V0 = 0

trays Fn + Ln–1 + Vn+1 – Ln – SLn – Vn – SVn = 0 nÎ[1, N]

reboiler LN – LN+1 – VN+1 = 0 (17)

where SL0 = D and LN+1 = B

– component material balance

V1yk 1 – (L0 + SL0) xk 0 – V0yk 0 = 0

Fnzk n + Ln-1xk n-1 + Vn+1yk n+1 – (Ln + SLn)xk n – (Vn + SVn)yk n = 0 nÎ[1, N] (18)

LNxk N – LN+1xk N+1 – VN+1yk N+1 = 0 kÎ[1, NK]

– energy balanceV H Q L SL H V H1 1 0 0 0 0 0 0

0V L V+ - + - =( )

F H L H V H Q L SL H V SV Hn n n n n n n n n n n nF L V L+ + + - + - +- - + +1 1 1 1

( ) ( ) n n NV = Î0 1[ , ] (19)

L H Q L H V HN N N n N N NL L V+ - - =+ + + + +1 1 1 1 1

0

– component molar fractions summation equation

x y n Nk nk

NK

k nk

NK=å =å Î +

= =1 1 0 1

1 1or [ , ] (20)



Figure 3. Equilibrium tray model

– equilibrium equation

yk n = Kk nxk n kÎ[1, NK] nÎ[0, N+1] (21)

To tal pro duced en tropy in col umn is:

d

d

irrL L V L VS

tL S SL S V S SL S SV S F SN N n n n n n n= + + + + -+ +1 1 0 0 0 0

( F ) - åå=

+

=

Q

Tn

nn

N

n

N

0

1

1(22)

In eq. (22), a mem ber for pro duced en tropy in exchangers, as in eq. (14), can be added.For the sys tem so lu tion of (2NK + 3)(N + 2) non-lin ear eqs. (17)-(21), the flows, com po si tions,feed streams enthalpies, ex changed heat on trays and the flows of all side prod ucts should begiven.

It can be as sumed for pres sure that it is con stant dur ing the whole col umn or that forev ery tray it is cal cu lated on the ba sis of known or as sumed pres sure drop. Num ber of vari ablesto be de ter mined, such as flows and com po si tions of va por and liq uid streams and tem per a tureon trays (in clud ing con denser and reboiler), is also (2NK + 3)(N + 2). There are four mainclasses of sep a ra tion col umns de scribed by these equa tions, de pend ing on spec i fi ca tions andthese are: ab sorber (desorber – strip per), reboiled ab sorber, rec ti fi ca tion col umn, and dis til la tion col umn. The sim plest case is ab sorber – col umn which does not have ei ther con denser orreboiler, so the num ber of equa tions and in de pend ent vari ables to be de ter mined is N(2NK + 3).

For dis til la tion col umn, flow V0 (0 if the con denser is to tal) and re flux rate are givenand that is why the val ues of flows L0 and V1 are fixed and in stead of them, heat quan ti ties to beex changed in the con denser and reboiler as un known are de ter mined.

At reboiled ab sorber, which has no con denser, heat as an un known quan tity, whichshould be in tro duced in the reboiler, is de ter mined in stead of flow V1, which is spec i fied. Rec ti -fi ca tion col umn has con denser only and the quan tity of heat, which should be re moved in thecon denser, is de ter mined in stead of V0, which is spec i fied (0 if con denser is a to tal one). Atreboiled ab sorber and rec ti fy ing col umn, the num ber of in de pend ent vari ables and equa tions is(2NK + 3)(N + 1). Some other spec i fi ca tions are pos si ble (e. g., some tem per a ture, com po si tionor some other prop erty) but they are rarely used.

Be cause the sys tem of sep a ra tion col umn math e mat i cal model equa tions is non-lin ear,it must be solved iteratively. There are a lot of meth ods and vari a tions, gen eral or spe cific forsome of spec i fied col umn classes.

At diabatic col umn, there is no re flux from the con denser, so the cal cu la tion of heatquan tity which needs to be re moved in the con denser is fixed by va por com po si tion from the topof the col umn. It is not con nected with solv ing col umn math e mat i cal model, i. e., it is not de ter -mined si mul ta neously with other in de pend ent vari ables but af ter they have been de ter mined.

Be cause of that, diabatic col umn cal cu la tion ap per tains to cal cu la tion class of reboiled ab sorber.

In or der to sim u late diabatic and anal o gous adi a batic col umn in the same method, oneof gen eral-global meth ods, is cho sen [26]. Mod i fi ca tions and up dates of this method al low sim -u la tion of cou pled col umns [27] and col umns with non-ideal trays de scribed by Murphy ef fi -ciency [28], and so lu tion of hard high non-ideal sep a ra tion prob lems [29, 30].

On the ba sis of this method, a pro gram for pro duced en tropy minimization given bythe eq. (22) is made. The fol low ing di rect meth ods for min i mum search ing are in cluded in thepro gram: Rosenbrock method of ro tat ing di rec tions [31], Hooke-Jeeves method [32], Powellmethod of con ju gated di rec tions [33], Sim plex method [34], and Com plex method [35].



Di rect meth ods are used be cause they only re quire knowl edge of func tion val ues and not the val ues of par tial de riv a tives which, in this case, can not be de ter mined an a lyt i cally andnu mer i cal cal cu la tion will be very time con sum ing. For ther mo dy namic prop er ties cal cu la tion – va por-liq uid equi lib rium con stants, enthalpies, and entropies, the Soave-Redlich-Kwong (SRK) [36] is used for real sys tems, and for ideal sys tems, the pro ce dures given in [17].

For bi nary col umn cal cu la tion, mi nor change to Ishii-Otto method is done be cause va -por tem per a ture from the top and tem per a ture from the first tray, re spec tively, is fixed by wanted dis til late com po si tion as dew point tem per a ture. In stead of it, the un known quan tity of heat to be re moved from the first tray is de ter mined. Ac cord ing to this, for bi nary col umn con trol vari -ables, op ti mi za tion quan ti ties of heat are Q2, Q3, ..., QN while for multicomponent col umn, theyare Q1, Q2, ..., QN.

Re sults and dis cus sion

For the new pro ce dure of pro duced en tropy minimization in diabatic dis til la tion, thecheck is done on fre quently used ex am ple of col umn for sep a ra tion of ben zene and toluol [5,17, 18, 25]. Feed mix ture with ben zene mo lar frac tion of 0.5 is in tro duced at the 12th tray witha flow of 1 mol/s at boil ing tem per a ture. Col umn is work ing at at mo spheric pres sure and has24 trays, to tal con denser, and par tial reboiler. Pres sure drop at trays is ne glected. Dis til lateflow is 0.5 mol/s and toluol mo lar frac tion in dis til late should not be higher than 0.05. The sys -tem is ob served as the ideal one and for ther mo dy namic val ues cal cu la tion, the pro ce dure de -

scribed in [17] based on ideal so lu tion model[37] is used, while data for com po nents aretaken from [38]. The most con ve nient op ti mi -za tion method for the ob served prob lem hascome out of the Sim plex method with 370 it er -a tions and 1090 cal cu la tions of func tion val ues fol low ing eq. (22). Min i mum value of pro -duced en tropy in diabatic col umn is 1.056 J/sKwhile at adi a batic col umn, it is 2.998 J/sK which is a sav ing of 64.77%. Other re sults areshown graph i cally as the com par i son of re -spected val ues of adi a batic and diabatic dis til -la tion. In fig. 4, where tem per a ture pro files areshown, uni form tem per a ture change at diabatic col umn trays can be seen, while the pro file inadi a batic one is S-shaped, i. e., the big gest tem -per a ture change is at trays at the end of the col -umn.

In adi a batic col umn va por and liq uid phases, flows are al most con stant in rec ti fy ing and strip ping sec tions, re spec tively, while in diabatic col -umn, these flows are the big gest at feed tray and they are uni formly de creased to ward the end ofthe col umn as it can be seen in figs. 5 and 6. Figure 7 shows op ti mal pro file of heat ex changed indiabatic col umn to gether with heat ex changed in con denser and reboiler of adi a batic col umn.Fig ure 8 gives pro duced en tropy pro files cal cu lated by eqs. (10)-(12) for both col umns. Pro -duced en tropy through trays is more uni formly dis trib uted at diabatic dis til la tion.



Fig ure 4. Tem per a ture pro file

Dur ing minimization, pre vi ous notes are con firmed that goal func tion given by eq.(13) – for con sid ered ex am ple eq. (22) is re duced to eq. (13) – has many lo cal min i mums withvery close val ues. Since only di rect op ti mi za tion meth ods have been used, there is a need to re -peat minimization pro ce dure sev eral times with dif fer ent ini tial val ues of con trol vari ables in or -der to en sure that ob tained so lu tion is as close as pos si ble to the global min i mum.

Con clu sions

The new pro ce dure for di rect nu mer i cal minimization of en tropy pro duc tion indiabatic col umn with in de pend ent heat exchangers at the trays is pro posed. The back bone of this



Figure 5. Vapor flow profile Figure 6. Liquid flow profile

Figure 7. Heat profile Figure 8. Total produced entropy profile

pro ce dure is the use of ex changed heat quan ti ties as con trol vari ables for op ti mi za tion. This ap -proach en ables the cal cu la tion of tem per a tures on the trays needed for cal cu la tion of goal func -tion val ues – pro duced en tropy to be done by solv ing math e mat i cal model of col umn. For thesepur poses, global Ishii-Otto method based on to tal linearization of model equa tions, par tic u larlyNew ton-Raphson tech nique, is used. For minimization, 5 well known di rect op ti mi za tion meth -ods, i. e., meth ods re quir ing only the knowl edge of goal func tion val ues and not val ues of the de -riv a tives, are used.

In the first step, the pro ce dure is suc cess fully ap plied to the ex am ple fre quently usedin lit er a ture in the col umn for sep a ra tion of ben zene and toluol mix ture giv ing ex pected re sultssim i lar to those in lit er a ture thus con firm ing the pos si bil ity of ap pli ca tion to bi nary col umns.

Ref er ences

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Chem i cal En gi neer ing, 14 (1974), 1, pp. 18-27[3] Kai ser, V., Gourlia, J. P., The Ideal Col umn Con cept: Ap ply ing Exergy to Dis til la tion, Chem i cal En gi -

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No men cla ture

A – heat transfer area, [m2]B – bottom products flow, [mol·s–1]D – distillate flow, [mol·s–1]F – feed mixture flow, [mol·s–1]H – enthalpy, [Jmol–1]K – vapor-liquid equilibrium constant, [–]k – component record number, [–]L – liquid streams flow, [mol·s–1]N – total trays number, [–]NK – total component number, [–]n – tray record number, [–]nF – feed tray record number, [–]Q – flow of exchanged heat, [Js–1]S – entropy, [Jmol–1K–1]dSirr/dt – total entropy production, [Js–1K–1]SL – liquid side stream flow, [mol·s–1]SV – vapor side stream flow, [mol·s–1]T – temperature, [K]V – vapor streams flow, [mol·s–1]XHX – heat transfer driving force in heat

– exchanger, [K–1]x – component molar fraction in liquid

– stream, [–]

y – component molar fraction in vapor– stream, [–]

z – component molar fraction in feed– mixture, [–]

Greek let ters

d – film thickness, [m]l – heat conductivity, [Wm–1K–1]

Su per scripts

B – bottomD – distillateF – feed mixtureL – liquidV – vapor

Sub scripts

k – component numbern – tray numberN – total tray number0 – condenser1 – first tray

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Paper submitted: December 9, 2009Paper revised: December 19, 2009Paper accepted: January 12, 2010



A new approach to entropy production minimization in diabatic distillation column with trays

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